Homological Arbitrage Overview

• Homological arbitrage is defined as a global market phenomenon where a consistent gain system is not generated by any price process due to nontrivial cohomology classes.
• It employs cohomological and multiplicative holonomy analyses on categorical filtrations to reveal arbitrage opportunities that local pricing models fail to detect.
• The framework integrates geometric and topological methods, linking loop effects, curvature, and global obstructions to the absence of traditional arbitrage.

Homological arbitrage is a class of arbitrage notions in which the obstruction to no-arbitrage is global rather than local. In the most explicit recent formalization, it arises in a cohomological martingale theory on categorical filtrations, where a gain system can be globally consistent yet fail to be generated by any price process; this failure is represented by a nontrivial first cohomology class (Adachi, 2 May 2026). Closely related work develops a multiplicative loop invariant, or holonomy, whose nontriviality yields “Aharonov–Bohm” arbitrage generated by loop effects rather than by any individual transition (Adachi, 12 Apr 2026). More broadly, the topic sits within a geometric and topological re-reading of arbitrage in which closed loops, curvature, failure of global potentials, and boundary obstructions replace purely local mispricing as the primary diagnostic objects (Meister, 18 Feb 2026, 0908.3043).

1. Terminological scope and conceptual core

In "Martingale Cohomology, Holonomy, and Homological Arbitrage" (Adachi, 2 May 2026), homological arbitrage is defined for a closed gain system $a\in Z_\beta^1$ that is not exact: $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$ The financial meaning is precise: the market admits a gain that is globally consistent but cannot be written as the gain generated by any price system. The paper explicitly identifies this with the analogue of a closed but non-exact $1$-form.

A related but distinct formulation appears in "Aharanov-Bohm Type Arbitrage and Homological Obstructions in Financial Markets" (Adachi, 12 Apr 2026). There the central object is not an additive cohomology class but a multiplicative distortion and its holonomy around loops. Arbitrage occurs when there exists a loop $\gamma$ such that

$\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$

The gain is therefore loop-level and global, even when each individual arrow appears locally innocuous.

The broader literature uses homological or cohomological language with different degrees of formality. "Caratheodory, Finite Resources and the Geometry of Arbitrage" states explicitly that it does not develop formal homology or cohomology theory in a rigorous algebraic-topology sense, even though it analyzes closed loops, cycles, global potentials, and nonzero circulation (Meister, 18 Feb 2026). "Gauge Invariance, Geometry and Arbitrage" does not use homology language, but it identifies arbitrage with the curvature of a gauge connection and path dependence with nontrivial holonomy (0908.3043).

Framework	Core object	Arbitrage signal
Homological arbitrage (Adachi, 2 May 2026)	$H_\beta^1$	$[a]\neq 0$
AB arbitrage (Adachi, 12 Apr 2026)	$\mathrm{Hol}(\gamma)$	$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$
Geometric/topological arbitrage (Meister, 18 Feb 2026, 0908.3043)	Loop circulation or curvature	Failure of global path-independence

2. Categorical filtrations and distorted martingale transport

The common formal setting in the cohomological and AB-arbitrage papers is a categorical filtration

$F:\mathcal T^{op}\to \mathrm{Prob},$

where $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$0 is a small category and $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$1 is the category of probability spaces and null-preserving measurable maps (Adachi, 2 May 2026, Adachi, 12 Apr 2026). Time is therefore not assumed to be a linear order; it may branch, merge, and contain loops.

The associated conditional expectation functor is

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$2

sending a probability space to its $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$3-space and a morphism $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$4 to a conditional expectation operator characterized by

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$5

This yields the composite

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$6

The central defect term is the distortion

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$7

for an arrow $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$8 in $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$9 (Adachi, 12 Apr 2026). In the cohomological formulation the same object appears as the density factor

$1$0

which measures the failure of conditional expectation to preserve constants under non-measure-preserving transport (Adachi, 2 May 2026). If $1$1 is measure-preserving, then $1$2; otherwise the deviation is multiplicative.

This distortion enters the martingale law. An $1$3-martingale is a family $1$4, with $1$5, such that for every arrow $1$6,

$1$7

When all morphisms are measure-preserving, $1$8, and the condition reduces to the classical martingale condition. The distortion therefore functions as the correction term that records nontrivial transport through the filtration.

The multiplicative consistency relation is

$1$9

for composable arrows $\gamma$0 and $\gamma$1 (Adachi, 12 Apr 2026). This is the cocycle-type relation that makes loop effects possible.

3. Cohomological normalization and the $\gamma$2-gauge

The formal cohomological theory is built on the nerve $\gamma$3, whose simplices

$\gamma$4

index simplicial cochains (Adachi, 2 May 2026). At degree zero,

$\gamma$5

so a $\gamma$6-cochain is an adapted process $\gamma$7.

The first coboundary-like operator is defined by

$\gamma$8

and the paper proves that

$\gamma$9

Martingales are thus exactly the $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$0-cocycles of the corrected theory.

A direct extension of this construction to higher degrees fails. The naive coboundary

$\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$1

does not satisfy the cochain identity: $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$2 in general, because the density operators $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$3 introduce multiplicative distortions. The paper exhibits a nonzero defect term $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$4, so the naive system is twisted rather than a genuine cochain complex.

The normalization procedure that removes this obstruction is the $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$5-gauge. For each simplex $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$6, the measures are recursively modified by

$\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$7

This produces a new diagram $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$8 in the subcategory $\mu_{t_0}\bigl(\mathrm{Hol}(\gamma)>1\bigr)>0.$9 of measure-preserving maps, with

$H_\beta^1$0

The corresponding coboundaries

$H_\beta^1$1

then satisfy

$H_\beta^1$2

The $H_\beta^1$3-gauge is therefore not an auxiliary convenience but the mechanism that converts a density-twisted martingale system into an honest cochain complex (Adachi, 2 May 2026).

4. Gain systems, first cohomology, and the formal definition

Once the $H_\beta^1$4-complex exists,

$H_\beta^1$5

its components acquire a direct financial interpretation (Adachi, 2 May 2026). The paper identifies $H_\beta^1$6 with price systems, $H_\beta^1$7 with martingales or consistent price systems, $H_\beta^1$8 with gain systems, $H_\beta^1$9 with gains generated by price systems, and

$[a]\neq 0$0

with consistent gains not arising from any price process.

A $[a]\neq 0$1-cochain $[a]\neq 0$2 assigns a gain to each arrow. Its cocycle law for composable arrows $[a]\neq 0$3 is

$[a]\neq 0$4

The later gain must be transported back by conditional expectation before being added to the earlier one. Exact gain systems are those of the form

$[a]\neq 0$5

for some price system $[a]\neq 0$6; these are gains produced by a potential.

Homological arbitrage is defined exactly at the gap between consistency and potentiality. A closed gain system $[a]\neq 0$7 exhibits homological arbitrage if it is not exact: $[a]\neq 0$8 The market then has a consistent gain system that no price process generates.

The paper illustrates the idea with loop examples. For a category generated by

$[a]\neq 0$9

with trivial measure transport, a $\mathrm{Hol}(\gamma)$0-cochain satisfying $\mathrm{Hol}(\gamma)$1 and $\mathrm{Hol}(\gamma)$2 has additive loop gain $\mathrm{Hol}(\gamma)$3. If $\mathrm{Hol}(\gamma)$4 is exact, this cancels to zero. By contrast, a loop $\mathrm{Hol}(\gamma)$5 with a gain system $\mathrm{Hol}(\gamma)$6 satisfying

$\mathrm{Hol}(\gamma)$7

is shown to be closed but not exact, with nonzero cohomological holonomy; this is the paper’s explicit example of intrinsic loop-level arbitrage (Adachi, 2 May 2026).

5. Holonomy, loops, and executable Aharonov–Bohm arbitrage

The cohomological paper introduces an additive holonomy for a closed gain system $\mathrm{Hol}(\gamma)$8 on a loop

$\mathrm{Hol}(\gamma)$9

subject to the loop-balance condition

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$0

(Adachi, 2 May 2026). The recursion is

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$1

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$2

with

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$3

This is an observable total loop gain.

For exact gain systems $\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$4, additive holonomy reduces to a transport defect: $\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$5 The associated transport defect space is

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$6

where $\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$7. The cohomological holonomy is then defined by

$\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$8

and depends only on the cohomology class $\mu_{t_0}(\mathrm{Hol}(\gamma)>1)>0$9. The purpose of this quotient is to remove loop effects that arise only from transport of a price system.

The multiplicative theory in (Adachi, 12 Apr 2026) parallels this construction with distortion rather than additive gain. For a loop

$F:\mathcal T^{op}\to \mathrm{Prob},$0

holonomy is defined recursively by

$F:\mathcal T^{op}\to \mathrm{Prob},$1

$F:\mathcal T^{op}\to \mathrm{Prob},$2

and

$F:\mathcal T^{op}\to \mathrm{Prob},$3

If $F:\mathcal T^{op}\to \mathrm{Prob},$4 along every arrow in the loop, then $F:\mathcal T^{op}\to \mathrm{Prob},$5. Nontrivial holonomy therefore records a global inconsistency invisible at the level of individual transitions.

AB arbitrage occurs when there exists a loop $F:\mathcal T^{op}\to \mathrm{Prob},$6 such that

$F:\mathcal T^{op}\to \mathrm{Prob},$7

The paper emphasizes the contrast with standard local arbitrage: there may be no detectable local anomaly on any single arrow, yet the loop product is nontrivial and yields gain (Adachi, 12 Apr 2026).

6. Admissibility, broader geometric programs, and interpretive limits

The multiplicative holonomy theory does not equate every categorical loop with an executable trading strategy. It introduces admissibility conditions: observability of $F:\mathcal T^{op}\to \mathrm{Prob},$8 at the base time, executability of each arrow as an available market transaction, composability of the sequence, self-financing, and reverse executability of $F:\mathcal T^{op}\to \mathrm{Prob},$9 when needed (Adachi, 12 Apr 2026). Under these conditions, one may define

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$00

with terminal wealth

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$01

The stated proposition is that

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$02

with strict inequality on

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$03

Thus, if

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$04

nontrivial holonomy yields a predictable self-financing trading strategy.

This executable-loop perspective aligns with a broader geometric literature on global arbitrage. "Caratheodory, Finite Resources and the Geometry of Arbitrage" distinguishes instantaneous non-arbitrage, expressed as local integrability of the wealth-change $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$05-form $$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$06, from general arbitrage, understood as failure of global path-independence (Meister, 18 Feb 2026). Its Boyling counterexample shows that a market may be locally consistent everywhere and still permit a closed-loop arbitrage because local potentials do not glue into a global one. The paper summarizes the hierarchy as

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$07

for instantaneous non-arbitrage, versus global inconsistency enabling “long-path” cycles with profit. It also states explicitly that its homological language is metaphorical: it does not explicitly define homology groups, cohomology groups, coboundaries, de Rham classes, Betti numbers, or chain complexes.

A different geometric program appears in "Gauge Invariance, Geometry and Arbitrage" (0908.3043). There arbitrage is encoded as a gauge connection whose curvature vanishes if and only if there is no arbitrage. The path-dependent pricing rule

$$a\notin \operatorname{im}(\delta_\beta^1), \qquad\text{equivalently}\qquad [a]\neq 0\in H_\beta^1.$$08

shows that nonzero curvature generates dependence on the self-financing path. This is not a cochain-complex construction, but it provides a closely related interpretation of arbitrage as a global obstruction to path independence.

The operator-theoretic study "A Complete Spectral Analysis of the CEV Operator with Applications to Arbitrage" does not use the term homological arbitrage, but it ties arbitrage-related phenomena to boundary behavior, self-adjoint extensions, positive harmonic functions, and strict-local-martingale regimes (Beretta et al., 21 May 2026). This suggests a broader structural theme: arbitrage can be encoded by global data that are invisible to strictly local diagnostics. A plausible implication is that “homological arbitrage” now names a family of approaches in which no-arbitrage is tested against global compatibility conditions—cohomology classes, loop holonomies, curvature, or boundary-sensitive spectral states—rather than against local consistency alone.